Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem

Lin Chen; Hui-Sheng Ding; Benniao Li; Jianghua Ye

doi:10.1515/ans-2022-0079

Article Open Access

Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem

Lin Chen , Hui-Sheng Ding , Benniao Li and Jianghua Ye

Published/Copyright: July 9, 2023

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From the journal Advanced Nonlinear Studies Volume 23 Issue 1

Abstract

In this article, we consider the following Schrödinger-Poisson problem:

− ε 2 Δ u + V ( y ) u + Φ ( y ) u = ∣ u ∣ p − 1 u , y ∈ R 3 , − Δ Φ ( y ) = u 2 , y ∈ R 3 ,

where ε > 0 is a small parameter, 1 < p < 5 , and V ( y ) is a potential function. We construct multi-peak solution concentrating at the critical points of V ( y ) through the Lyapunov-Schmidt reduction method. Moreover, by using blow-up analysis and local Pohozaev identities, we prove that the multi-peak solution we construct is non-degenerate. To our knowledge, it seems be the first non-degeneracy result on the Schödinger-Poisson system.

Keywords: multi-peak solution; non-degeneracy; Schrödinger-Poisson system

MSC 2010: 35J15; 35J20

1 Introduction

From the point view of Quantum Mechanics, the Schrödinger-Poisson system describes the mutual interactions of many particles [29]. The behaviour of a single particle of mass m > 0 can be described by the linear Schrödinger equation:

i ℏ ∂ W ∂ t = − ℏ 2 m Δ W + a ( y ) W + Φ ( y , t ) W , y ∈ R 3 , t ∈ R ,

where i is the imaginary unit, Δ is the Laplacian operator, ℏ is the Planck constant, and Φ : R 3 × R → R . In contrast to the single-particle case, in the presence of many particles, we can model the effect of mutual interactions by introducing a nonlinear term. Then one leads to a non-linear equation of the form

i ℏ ∂ W ∂ t = − ℏ 2 m Δ W + a ( y ) W + Φ ( y , t ) W − ∣ W ∣ p − 1 W , y ∈ R 3 , t ∈ R ,

with 1 < p < 5 . If the particle moves in its own gravitational field, which is generated by the probability density of the particle via the Newtonian field equation, then the potential

Φ ( y , t ) = 1 4 π ∫ R 3 ∣ W ( z , t ) ∣ 2 ∣ y − z ∣ d z

is a solution of the Poisson equation:

− Δ y Φ = ∣ W ∣ 2 .

We look for standing-wave solutions with the form W ( y , t ) = u ( y ) e i ϖ t , ϖ > 0 , y ∈ R 3 , t ∈ R , then the system becomes

(1.1) − ε 2 Δ u + V ( y ) u + Φ ( y ) u = ∣ u ∣ p − 1 u , y ∈ R 3 , − Δ Φ ( y ) = u 2 , y ∈ R 3 ,

where ε 2 = ℏ 2 m and V ( y ) = a ( y ) + ℏ ϖ . System (1.1) also described the interaction of a charge particle with an electromagnetic field [6,10,11].

Mathematically, many results on existence of solutions for system (1.1) are established. For a fixed constant ε > 0 , these works [10,11,25] studied the existence of solutions for system (1.1) when the function V ( y ) is a constant-valued function, while the existence of solutions of Schrödinger-Poisson system with non-constant potential functions V ( y ) was taken into consideration in [5]. On the other hand, there are a lot of results on the existence of solutions for the singularly perturbed problem (1.1), that is, ε is a small parameter. In [24], Ruiz proved that for 1 < p < 11 7 , system (1.1) possesses a family of solutions concentrating around a sphere as ε → 0 when V = 1 . In [16,18], the authors investigated the existence of solutions to the Schrödinger-Poisson problem and the solutions concentrate on the sphere when the weight function is radially symmetric. The single-peak solution of the Schrödinger-Poisson problem was studied in [17] and the cluster solutions for system (1.1) was constructed in [26]. More results about the Schrödinger-Poisson system can also refer to [1,2,4,7,13–16,21,27] and the references therein.

It is known that the non-degeneracy of solutions is also an important property in the theory of differential equations when one deals with the stability or instability of the solutions. On the other hand, the non-degenerate nature of the solutions can be used to prove the existence results of solutions through the famous Lyapunov-Schmidt reduction procedure. However, to the best of our knowledge, the non-degeneracy of solutions for the Schrödinger-Poisson problem has not been investigated. Here, we focus on the non-degenerate behaviour of a class of concentrating solutions to (1.1). For simplicity, we suppose V satisfies:

V ( y ) ∈ C 2 ( R 3 , R ) , 0 < V 0 ≤ V ( y ) ≤ V 1 ;
V ( y ) has m non-degenerate critical points p 1 , … , p m .

In this article, our main concern is the non-degeneracy of peak solutions for Schrödinger-Poisson systems with non-degenerate potentials. However, it is needed to note that there are some interesting results on the existence of peak solutions for related problems with degenerate potentials, see [22,23] and the references therein. Lu and Wei [22] studied concentrated positive bound states of nonlinear Schrödinger equations with totally degenerate potentials, and showed how exactly the total degeneracy of potentials can affect the existence and properties of solutions. Luo et al. [23] investigated the existence and uniqueness of normalized solutions for Bose-Einstein condensates with degenerate potentials.

We will use the unique ground state w of

− Δ w + w = w p , w > 0 , in R 3 , w ( 0 ) = max y ∈ R 3 w ( y ) , w ∈ H 1 ( R 3 )

to build up the approximate solution for system (1.1). As shown in [3,19], w ( y ) = w ( ∣ y ∣ ) satisfies

w ′ ( r ) < 0 , lim r → ∞ r e r w ( r ) = C > 0 , lim r → ∞ w ′ ( r ) w ( r ) = − 1 .

Moreover, w ( y ) is non-degenerate, that is,

Kerl ℒ = span ∂ w ∂ y i , i = 1 , 2 , 3 ,

where the operator is given by, for any φ ∈ H 1 ( R 3 ) ,

ℒ φ = − Δ φ + φ − p w p − 1 φ .

Fixing x ∈ R 3 , we denote

U ε , x ( y ) ≔ ( V ( x ) ) 1 p − 1 w V ( x ) ε ( y − x ) .

The existence of multi-peak solutions for (1.1) can be established following by the similar way with the work in [17], and the proof will be omitted here.

Theorem 1.1

If V ( y ) satisfies ( V 1 ) and ( V 2 ) , then there exist θ > 0 and ε 0 > 0 such that for any ε ∈ ( 0 , ε 0 ] , system (1.1) has a solution of the form

(1.2) u ε = ∑ j = 1 m U ε , x ε , j + ω ε

for some x ε , j ∈ B θ ( p j ) , and ‖ ω ε ‖ ε = O ε 5 2 , where ‖ u ‖ ε 2 ≔ ∫ R 3 ( ε 2 ∣ ∇ u ∣ 2 + V ( y ) u 2 ) d y .

Remark 1.2

The assumption on V ( y ) can be weaken in Theorem 1.1. In fact, the result holds when Condition ( V 2 ) is substituted by the degenerate condition, that is, there exists an even integer n ∈ [ 4 , ∞ ] such that p j is a degenerate local minimum( or maximum) of V ( y ) and

D k V ( p j ) = 0 , k = 1 , 2 , … , n − 1 ; j = 1 , 2 , … , m and D n V ( p j ) [ x ] = ∑ i = 1 3 a i , j x i n ,

where a i , j = ∂ n V ( p j ) ∂ x i n and a i , j > 0 (or a i , j < 0 ).

Next, we will study the non-degeneracy of solution u ε . Define

A ε v ≔ − ε 2 Δ v + ( V ( y ) + Φ u ε − p ∣ u ε ∣ p − 1 ) v + 1 2 π ∫ R 3 u ε ( z ) v ( z ) ∣ y − z ∣ d z u ε

for any v ∈ H 1 ( R 3 ) . Inspired by the non-degeneracy of solution for Schrödinger equation in [28], we have the following non-degeneracy result through using the blow-up analysis and the local Pohozaev identities:

Theorem 1.3

If V ( y ) satisfies ( V 1 ) and ( V 2 ) , v ε satisfies A ε v ε = 0 , then v ε = 0 for sufficiently small ε > 0 .

Remark 1.4

Theorems 1.1 and 1.3 hold when the dimension N satisfies 3 < N ≤ 6 .

Following with [8], we argued by contradiction. By the linearity of the operator A ε , we can assume that ‖ v ε ‖ L ∞ ( R 3 ) = 1 . For the estimates of v ε near the non-degenerate points, we can use the blow-up analysis and local Pohozaev identities, while we will use the comparison principle to get the estimates away from these points.

Before closing this section, we will point out the difficulties in this article. Compared with the classical Schrödinger equation, there is a more non-local term in Schrödinger-Poisson system (1.1). The non-local term brings in much more difficulties when we study the non-degeneracy of the multi-peak solution with the form (1.2). For example, there exist two double-volume integrals in the local Pohozaev identities (2.5). To deal with these two double-volume integrals, we need to estimate accurately and skilfully. On the other hand, much more difficulties are brought by the non-local term as well when we estimate ∣ v ε ∣ and ∣ ∇ v ε ∣ , where v ε satisfies (3.2).

Our article is organized as follows. In Section 2, we will carry out some basic results to apply in the proof of the main theorem further. In Section 3, we will give the proof of Theorem 1.3. In the sequel, we will use C and σ to denote various generic positive constants and small positive constants, respectively.

2 Preliminaries

In this section, we will give some useful results to apply further in the proof. For every u ∈ H 1 ( R 3 ) , it follows from Lax-Milgram theorem that there exists a unique Φ = Φ u ∈ D 1 , 2 ( R 3 ) such that − Δ Φ = u 2 , where

(2.1) Φ u ( x ) = 1 4 π ∫ R 3 u 2 ( z ) ∣ x − z ∣ d z .

The properties of Φ u are as follows, which can be proved similar to Lemma 2.1 [9].

Lemma 2.1

For any u , v ∈ H 1 ( R 3 ) , we have

‖ Φ u ‖ D 1 , 2 ( R 3 ) ≤ 1 S 1 2 ‖ u ‖ L 12 5 ( R 3 ) 2 ,
‖ Φ u − Φ v ‖ D 1 , 2 ( R 3 ) ≤ 1 S 1 2 ‖ u 2 − v 2 ‖ L 6 5 ( R 3 ) , where S ≔ inf u ∈ D 1 , 2 ( R 3 ) ‖ u ‖ L 6 ( R 3 ) = 1 ∫ R 3 ∣ ∇ u ∣ 2 d x .

Next, we give some important estimates. The following result can be found in Lemma 3.2 [12], which can be used later.

Lemma 2.2

For every α ∈ { 1 , … , N − 1 } and f : R N → R such that ( 1 + ∣ y ∣ α + 1 ) f ∈ L 1 ( R N ) ∩ L ∞ ( R N ) , set

Ψ α [ f ] ( y ) = ∫ R N f ( z ) ∣ y − z ∣ α d z .

Then there exist two positive constants C ( α , f ) and C ′ ( α , f ) such that

Ψ α [ f ] ( y ) − C ( α , f ) ∣ y ∣ α ≤ C ′ ( α , f ) ∣ y ∣ α + 1 , y ≠ 0 .

Corollary 2.3

Suppose f satisfies the condition in Lemma 2.2, then there exists a positive constant C ″ ( α , f ) such that

(2.2) ∣ Ψ α [ f ] ( y ) ∣ ≤ C ″ ( α , f ) ∣ y ∣ α , y ≠ 0 .

Proof

When ∣ y ∣ ≥ 1 , (2.2) holds obviously by Lemma 2.2.

On the other hand, when ∣ y ∣ ≤ 1 and y ≠ 0 , we have

(2.3) ∫ B ∣ y ∣ ( y ) f ( z ) ∣ y − z ∣ α d z ≤ ‖ f ‖ L ∞ ( R N ) ∫ B ∣ y ∣ ( 0 ) 1 ∣ x ∣ α d x ≤ C ‖ f ‖ L ∞ ( R N ) 1 ∣ y ∣ α

and

(2.4) ∫ R N \ B ∣ y ∣ ( y ) f ( z ) ∣ y − z ∣ α d z ≤ 1 ∣ y ∣ α ∫ R N ∣ f ( z ) ∣ d z = ‖ f ‖ L 1 ( R N ) 1 ∣ y ∣ α .

It follows from (2.3) and (2.4) that (2.2) holds.□

Lemma 2.4

(Pohozaev identities) If u ε is a solution of

− ε 2 Δ u ε + V ( y ) u ε + Φ u ε u ε = ∣ u ε ∣ p − 1 u ε , in R 3

and v ε is a solution of

− ε 2 Δ v ε + V ( y ) v ε + Φ u ε v ε + 1 2 π u ε ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z = p ∣ u ε ∣ p − 1 v ε ,

then it holds that, for any Ω ⊂ R 3 ,

(2.5) ∫ Ω ∂ V ( y ) ∂ y i u ε v ε d y = − ε 2 ∫ ∂ Ω ∂ u ε ∂ n ∂ v ε ∂ y i + ∂ v ε ∂ n ∂ u ε ∂ y i d S + ε 2 ∫ ∂ Ω ∇ u ε ∇ v ε n i d S − ∫ ∂ Ω ( ∣ u ε ∣ p − 1 u ε − V ( y ) u ε ) v ε n i d S − 1 4 π ∫ Ω ∫ R 3 ( z i − y i ) u ε 2 ( z ) ∣ y − z ∣ 3 d z u ε v ε d y + ∫ ∂ Ω Φ u ε u ε v ε n i d S − 1 4 π ∫ Ω ∫ R 3 ( z i − y i ) u ε ( z ) v ε ( z ) ∣ y − z ∣ 3 d z u ε 2 d y + 1 4 π ∫ ∂ Ω ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z u ε 2 n i d S ,

where n is the outward unit normal of ∂ Ω and n = ( n 1 , n 2 , n 3 ) .

Proof

Since

( − ε 2 Δ u ε + V ( y ) u ε + Φ u ε u ε ) ∂ v ε ∂ y i = ∣ u ε ∣ p − 1 u ε ∂ v ε ∂ y i

and

− ε 2 Δ v ε + V ( y ) v ε + Φ u ε v ε + 1 2 π u ε ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z ∂ u ε ∂ y i = p ∣ u ε ∣ p − 1 v ε ∂ u ε ∂ y i ,

by the divergence theorem, we have

(2.6) − ε 2 ∫ Ω Δ u ε ∂ v ε ∂ y i + Δ v ε ∂ u ε ∂ y i d y = − ∫ Ω V ( y ) u ε ∂ v ε ∂ y i + v ε ∂ u ε ∂ y i d y − ∫ Ω Φ u ε u ε ∂ v ε ∂ y i + v ε ∂ u ε ∂ y i d y + ∫ Ω ∣ u ε ∣ p − 1 u ε ∂ v ε ∂ y i + p ∣ u ε ∣ p − 1 v ε ∂ u ε ∂ y i d y − 1 2 π ∫ Ω ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z u ε ( y ) ∂ u ε ∂ y i d y = ∫ Ω ∂ V ( y ) ∂ y i u ε v ε d y − ∫ ∂ Ω V ( y ) u ε v ε n i d S + ∫ Ω ∂ Φ u ε ∂ y i u ε v ε d y − ∫ ∂ Ω Φ u ε u ε v ε n i d S + ∫ ∂ Ω ∣ u ε ∣ p − 1 u ε v ε n i d S + 1 4 π ∫ Ω ∫ R 3 ( z i − y i ) u ε ( z ) v ε ( z ) ∣ y − z ∣ 3 d z u ε 2 d y − 1 4 π ∫ ∂ Ω ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z u ε 2 n i d S = ∫ ∂ Ω ( ∣ u ε ∣ p − 1 u ε − V ( y ) u ε ) v ε n i d S + ∫ Ω ∂ V ( y ) ∂ y i u ε v ε d y − ∫ ∂ Ω Φ u ε u ε v ε n i d S + 1 4 π ∫ Ω ∫ R 3 ( z i − y i ) u ε 2 ( z ) ∣ y − z ∣ 3 d z u ε v ε d y + 1 4 π ∫ Ω ∫ R 3 ( z i − y i ) u ε ( z ) v ε ( z ) ∣ y − z ∣ 3 d z u ε 2 d y − 1 4 π ∫ ∂ Ω ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z u ε 2 n i d S .

By using integration by parts, we obtain

∫ Ω Δ u ε ∂ v ε ∂ y i d y = − ∑ l = 1 3 ∫ Ω ∂ u ε ∂ y l ∂ 2 v ε ∂ y i ∂ y l d y + ∫ ∂ Ω ∂ u ε ∂ n ∂ v ε ∂ y i d S .

Similarly, we have

∫ Ω Δ v ε ∂ u ε ∂ y i d y = − ∑ l = 1 3 ∫ Ω ∂ v ε ∂ y l ∂ 2 u ε ∂ y i ∂ y l d y + ∫ ∂ Ω ∂ v ε ∂ n ∂ u ε ∂ y i d S .

Thus,

(2.7) − ε 2 ∫ Ω Δ u ε ∂ v ε ∂ y i + Δ v ε ∂ u ε ∂ y i d y = ε 2 ∑ l = 1 3 ∫ Ω ∂ ∂ y i ∂ u ε ∂ v ε ∂ y l ∂ y l d y − ε 2 ∫ ∂ Ω ∂ u ε ∂ n ∂ v ε ∂ y i + ∂ v ε ∂ n ∂ u ε ∂ y i d S = ε 2 ∫ ∂ Ω ∇ u ε ∇ v ε n i d S − ε 2 ∫ ∂ Ω ∂ u ε ∂ n ∂ v ε ∂ y i + ∂ v ε ∂ n ∂ u ε ∂ y i d S .

The result follows from (2.6) and (2.7).□

3 The non-degeneracy result

In this section, we will prove that the solution with the form (1.2) of system (1.1) is non-degenerate. By the similar way with [8], we can prove the following results, which will be used later.

Lemma 3.1

Assume that u ε is a multi-peak solution of (1.1) with the form (1.2). Then there exist λ ∈ ( 0 , V 0 ) and C > 0 , such that

‖ ω ε ‖ L ∞ ( R 3 ) = o ( 1 ) , ∣ ω ε ( y ) ∣ ≤ C ∑ j = 1 m e − λ ∣ y − x ε , j ∣ ε , for y ∈ R 3 ,

and

∣ ∇ ω ε ( y ) ∣ ≤ C e − λ d 4 ε , for y ∈ ∂ B d ( x ε , j ) ,

where 0 < d ≤ 1 4 min h ≠ j ∣ p h − p j ∣ .

According to Lemma 3.1, we have

(3.1) ∣ u ε ( y ) ∣ ≤ C ∑ j = 1 m e − λ ∣ y − x ε , j ∣ ε , for y ∈ R 3

and

∣ ∇ u ε ( y ) ∣ ≤ C e − λ d 4 ε , for y ∈ ∂ B d ( x ε , j ) .

Proposition 3.2

Assume that u ε is a solution of (1.1) with the form (1.2) concentrating at p 1 , p 2 , … , p m , which are different non-degenerate points of V ( y ) . Then, there hold

∣ x ε , j − p j ∣ = O ( ε 2 ) , j = 1 , 2 , … , m .

Now, we prove Theorem 1.3 by contradiction. Due to the linear property of the operator A ε , we can suppose that there are ε n → 0 satisfying ‖ v ε n ‖ L ∞ ( R 3 ) = 1 and A ε n v ε n = 0 . For simplicity, we drop the subscript n . In order to get a contradiction, we want to prove ‖ v ε ( y ) ‖ L ∞ ( R 3 ) < 1 2 when ε is small enough. At first, we will prove that v ε decays exponentially away from the concentrating points so that ∣ v ε ( y ) ∣ < 1 2 for y ∈ R 3 \ ⋃ j = 1 m B ε R ( x ε , j ) , where R > 0 large enough. On the other hand, we study the local behaviours of v ε near each concentrating point through the blow-up analysis

v ε , j ( y ) ≔ v ε ( ε y + x ε , j ) , j = 1 , 2 , … , m .

We will prove v ε , j → 0 in C 1 ( B R ( 0 ) ) as ε → 0 , by using local Pohozaev identities.

Lemma 3.3

We have

‖ v ε ‖ ε = O ε 3 2 .

Proof

Because v ε satisfies

(3.2) − ε 2 Δ v ε + V ( y ) v ε = p ∣ u ε ∣ p − 1 v ε − Φ u ε v ε − 1 2 π u ε ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z ,

we have

(3.3) ‖ v ε ‖ ε 2 = p ∫ R 3 ∣ u ε ∣ p − 1 v ε 2 d y − ∫ R 3 Φ u ε v ε 2 d y − 1 2 π ∫ R 3 ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z u ε v ε d y .

It follows from (3.1) and ‖ v ε ‖ L ∞ ( R 3 ) = 1 that

(3.4) ∫ R 3 ∣ u ε ∣ p − 1 v ε 2 d y ≤ ∫ R 3 ∣ u ε ∣ p − 1 d y ≤ C ∫ R 3 ∑ j = 1 m e − λ ∣ y − x ε , j ∣ ε p − 1 d y = O ( ε 3 ) .

According to Hardy-Littlewood-Sobolev inequality and Hölder inequality, we know

(3.5) ∫ R 3 ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z u ε v ε d y ≤ C ‖ u ε v ε ‖ L 6 5 ( R 3 ) 2 ≤ C ‖ u ε ‖ L 12 5 ( R 3 ) 2 ‖ v ε ‖ L 12 5 ( R 3 ) 2 ≤ C ε 2 ‖ v ε ‖ ε 2

and

(3.6) ∫ R 3 Φ u ε v ε 2 d y ≤ C ‖ u ε 2 ‖ L 6 5 ( R 3 ) ‖ v ε 2 ‖ L 6 5 ( R 3 ) = C ‖ u ε ‖ L 12 5 ( R 3 ) 2 ‖ v ε ‖ L 12 5 ( R 3 ) 2 ≤ C ε 2 ‖ v ε ‖ ε 2 .

By (3.3)–(3.6), we obtain

‖ v ε ‖ ε 2 ≤ C ( ε 3 + ε 2 ‖ v ε ‖ ε 2 ) ,

which implies ‖ v ε ‖ ε = O ε 3 2 . □

The next lemma shows the estimate of ∣ v ε ∣ and ∣ ∇ v ε ∣ .

Lemma 3.4

There exist σ > 0 and C > 0 such that

(3.7) ∣ v ε ( y ) ∣ ≤ C ∑ j = 1 m e − σ ∣ y − x ε , j ∣ ε , y ∈ R 3

and

(3.8) ∣ ∇ v ε ( y ) ∣ ≤ C e − σ d 4 ε , y ∈ ∂ B d ( x ε , j ) ,

where 0 < d ≤ 1 4 min h ≠ j ∣ p h − p j ∣ .

Proof

By (3.1), for λ in Lemma 3.1, there exists R > 0 such that

V ( y ) + Φ u ε − p ∣ u ε ∣ p − 1 ≥ λ 2 , y ∈ R 3 \ ⋃ j = 1 m B ε R ( x ε , j ) .

When y ∈ Ω ε , 1 ≔ { y ∈ R 3 \ ⋃ j = 1 m B ε R ( x ε , j ) : v ε ( y ) ≥ 0 } , we have

− ε 2 Δ v ε + λ 2 v ε ≤ f ε ,

where f ε ( y ) ≔ − 1 2 π u ε ( y ) ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z .

Suppose ξ ε is the solution of equation,

− ε 2 Δ ξ ε + λ 2 ξ ε = f ε .

Define ξ ε ˜ ( y ) = ξ ε ( ε y ) , then ξ ε ˜ satisfies

− Δ ξ ε ˜ ( y ) + λ 2 ξ ε ˜ ( y ) = f ε ( ε y ) .

By Theorem 6.23 in [20], we have

ξ ε ˜ ( y ) = 1 4 π ∫ R 3 1 ∣ y − z ∣ e − λ ∣ y − z ∣ f ε ( ε z ) d z .

Therefore,

ξ ε ( y ) = ξ ε ˜ y ε = 1 4 π ∫ R 3 1 ∣ y ε − z ∣ e − λ y ε − z f ε ( ε z ) d z .

Next, we will give the estimates of ∣ ξ ε ( y ) ∣ . It follows from Hölder inequality that

(3.9) ∣ ξ ε ( y ) ∣ = 1 8 π 2 ∫ R 3 1 y ε − z e − λ y ε − z ∫ R 3 u ε ( x ) v ε ( x ) ∣ ε z − x ∣ d x u ε ( ε z ) d z ≤ 1 8 π 2 ε 2 ∫ R 3 e − λ ε ∣ y − z ∣ ∣ y − z ∣ ∫ R 3 ∣ u ε ( x ) ∣ ∣ v ε ( x ) ∣ ∣ z − x ∣ d x ∣ u ε ( z ) ∣ d z ≤ C ε 2 ∑ j = 1 m e − σ ε ∣ y − x ε , j ∣ ∫ R 3 1 ∣ y − z ∣ e − λ − σ ε ∣ y − z ∣ ( Φ u ε ( z ) Φ v ε ( z ) ) 1 2 d z ≤ C ε 2 ∑ j = 1 m e − σ ε ∣ y − x ε , j ∣ ∫ R 3 e − 6 ( λ − σ ) 5 ε ∣ y − z ∣ ∣ y − z ∣ 6 5 d z 5 6 ‖ Φ u ε ‖ L 6 ( R 3 ) 1 2 ‖ Φ v ε ‖ L 6 ( R 3 ) 1 2 ,

where σ ∈ ( 0 , λ ) . It is easy to verify that

(3.10) ∫ R 3 e − 6 ( λ − σ ) 5 ε ∣ y − z ∣ ∣ y − z ∣ 6 5 d z ≤ C ε 9 5 .

By Lemmas 2.1 and 3.3, we have

(3.11) ‖ Φ u ε ‖ L 6 ( R 3 ) 1 2 ≤ C ε 5 4 and ‖ Φ v ε ‖ L 6 ( R 3 ) 1 2 ≤ C ε 5 4 .

Thanks to (3.9)–(3.11), it holds

(3.12) ∣ ξ ε ( y ) ∣ ≤ C ε 2 ∑ j = 1 m e − σ ε ∣ y − x ε , j ∣ .

Denote that g ε ( y ) = v ε ( y ) − ξ ε ( y ) , then g ε satisfies that

− ε 2 Δ g ε + λ 2 g ε ≤ 0 , in Ω ε , 1 .

Let

T ε ( u ) = − ε 2 Δ u + λ 2 u .

By the direct computation, we can obtain

T ε ( e − σ ∣ y − x ε , j ∣ ε ) > 0 .

Because of (3.12) and ‖ v ε ‖ L ∞ ( R 3 ) = 1 , there exists M > 0 , such that

∣ g ε ∣ < M .

Denote

g ˜ ε ( y ) = M e σ R ∑ j = 1 m e − σ ∣ y − x ε , j ∣ ε − g ε ( y ) .

Thus,

T ε ( g ˜ ε ) > 0 in Ω ε , 1 .

Moreover, for y ∈ ∂ Ω ε , 1 , we have

g ˜ ε ( y ) > 0 .

By the comparison principle, we obtain that

g ˜ ε ( y ) ≥ 0 , for y ∈ Ω ε , 1 ,

which implies that

v ε ( y ) ≤ ∣ ξ ε ( y ) ∣ + M e σ R ∑ j = 1 m e − σ ∣ y − x ε , j ∣ ε ≤ C ∑ j = 1 m e − σ ∣ y − x ε , j ∣ ε , for y ∈ Ω ε , 1 .

Analogously, when y ∈ Ω ε , 2 ≔ { y ∈ R 3 \ ⋃ j = 1 m B ε R ( x ε , j ) : v ε ( y ) < 0 } , it holds

0 > v ε ( y ) ≥ − ∣ ξ ε ( y ) ∣ − M e σ R ∑ j = 1 m e − σ ∣ y − x ε , j ∣ ε ≥ − C ∑ j = 1 m e − σ ∣ y − x ε , j ∣ ε .

Therefore, we obtain

∣ v ε ( y ) ∣ ≤ C ∑ j = 1 m e − σ ε ∣ y − x ε , j ∣ , y ∈ R 3 \ ⋃ j = 1 m B ε R ( x ε , j ) .

For y ∈ ⋃ j = 1 m B ε R ( x ε , j ) , we have

∣ v ε ( y ) ∣ ≤ 1 ≤ e σ R ∑ j = 1 m e − σ ε ∣ y − x ε , j ∣ .

Thus, (3.7) holds.

Next, we prove (3.8). Because v ε satisfies

− Δ v ε = 1 ε 2 [ p ∣ u ε ∣ p − 1 v ε − V ( y ) v ε − Φ u ε v ε − 1 2 π u ε ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z ] ,

by L q -estimate, one has, for x ∈ ∂ B d ( x ε , j ) , q < 6 ,

(3.13) ‖ v ε ‖ W 2 , q B d 4 ( x ) ≤ C ε 2 p ∣ u ε ∣ p − 1 v ε − V v ε − Φ u ε v ε − 1 2 π u ε ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z L q B d 2 ( x ) + C ‖ v ε ‖ L q B d 2 ( x ) ≤ C ε 2 ‖ v ε ‖ L q B d 2 ( x ) + ‖ Φ u ε v ε ‖ L q B d 2 ( x ) + u ε ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z L q B d 2 ( x ) + C ‖ v ε ‖ L q B d 2 ( x ) ≤ C ε 2 ‖ v ε ‖ L q B d 2 ( x ) + ‖ Φ u ε v ε ‖ L q B d 2 ( x ) + u ε ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z L q B d 2 ( x ) .

When y ∈ B d 2 ( x ) , then d 2 ≤ ∣ y − x ε , j ∣ ≤ 3 d 2 . If ε is small enough, ∣ x ε , j − x ε , h ∣ ≥ 1 2 ∣ p j − p h ∣ ≥ 2 d , we obtain ∣ y − x ε , h ∣ ≥ ∣ x ε , j − x ε , h ∣ − ∣ y − x ε , j ∣ ≥ 2 d − 3 d 2 = d 2 , h ≠ j . Therefore,

(3.14) ‖ v ε ‖ L q B d 2 ( x ) = ∫ B d 2 ( x ) ∣ v ε ( y ) ∣ q d y 1 q ≤ C ∫ B d 2 ( x ) ∑ h = 1 m e − σ ε ∣ y − x ε , h ∣ q d y 1 q ≤ C ∫ B d 2 ( x ) ∑ h = 1 m e − σ d 2 ε q d y 1 q ≤ C e − σ d 2 ε .

By Hölder inequality and Lemma 2.1, we have

(3.15) ‖ Φ u ε v ε ‖ L q B d 2 ( x ) ≤ ‖ Φ u ε ‖ L 6 B d 2 ( x ) ‖ v ε ‖ L 6 q 6 − q B d 2 ( x ) ≤ C e − σ d 2 ε ‖ Φ u ε ‖ L 6 ( R 3 ) ≤ C e − σ d 2 ε

and

(3.16) u ε ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z L q B d 2 ( x ) ≤ ∫ B d 2 ( x ) ∣ ( Φ u ε Φ v ε ) 1 2 u ε ∣ q d y 1 q ≤ ‖ Φ u ε ‖ L 6 B d 2 ( x ) 1 2 ‖ Φ v ε ‖ L 6 B d 2 ( x ) 1 2 ‖ u ε ‖ L 6 q 6 − q B d 2 ( x ) ≤ C ‖ u ε ‖ L 6 q 6 − q B d 2 ( x ) ≤ C e − σ d 2 ε .

It follows from (3.13)–(3.16) that

(3.17) ‖ v ε ‖ W 2 , q B d 4 ( x ) ≤ C ε 2 e − σ d 2 ε ≤ C e − σ d 4 ε , for q < 6 .

Taking 3 < q < 6 , according to (3.17) and Sobolev embedding theorem, (3.8) holds.□

Now, we study the local behaviours of v ε near each concentrating point.

Lemma 3.5

We have

v ε , j ( y ) → ∑ i = 1 3 a i j ( V ( p j ) ) 1 p − 1 ∂ w ( V ( p j ) y ) ∂ y i i n C 1 ( B R ( 0 ) ) a s ε → 0 ,

for some constants a i j , i = 1 , 2 , 3 , j = 1 , 2 , … , m .

Proof

We notice that v ε , j satisfies the equation

− Δ v ε , j + V ( ε y + x ε , j ) v ε , j + ε 2 J ε , 1 = J ε , 2 v ε , j ,

where

J ε , 1 ( y ) ≔ 1 4 π ∫ R 3 u ε 2 ( ε z + x ε , j ) ∣ y − z ∣ d z v ε , j ( y ) + 1 2 π ∫ R 3 u ε ( ε z + x ε , j ) v ε , j ( z ) ∣ y − z ∣ d z u ε ( ε y + x ε , j )

and

J ε , 2 ( y ) ≔ p ∣ u ε ( ε y + x ε , j ) ∣ p − 1 .

Since ‖ v ε , j ‖ L ∞ ( R 3 ) = 1 , we have ∣ J ε , 1 ( y ) ∣ ≤ C . By Lemma 3.1, we obtain

u ε ( ε y + x ε , j ) = ∑ h = 1 m U ε , x ε , h ( ε y + x ε , j ) + ω ε ( ε y + x ε , j ) = ( V ( x ε , j ) ) 1 p − 1 w ( V ( x ε , j ) y ) + o ( 1 ) = ( V ( p j ) ) 1 p − 1 w ( V ( p j ) y ) + o ( 1 ) , for y ∈ B R ( 0 ) ,

which implies

J ε , 2 ( y ) = p V ( p j ) w p − 1 ( V ( p j ) y ) + o ( 1 ) .

According to the L q regular theorem and the Schauder estimate, we have v ε , j → v j in C 2 ( R 3 ) and v j satisfies

− Δ v j + V ( p j ) v j = p V ( p j ) w p − 1 ( V ( p j ) y ) v j .

It follows from the non-degeneracy of w that

v j = ∑ i = 1 3 a i j ( V ( p j ) ) 1 p − 1 ∂ w ( V ( p j ) y ) ∂ y i

for some constants a i j .□

Lemma 3.6

Let a i j be as in Lemma 3.5, then we have

a i j = 0 , i = 1 , 2 , 3 ; j = 1 , 2 , … , m .

Proof

We will use Pohozaev identities in Lemma 2.4 to prove by choosing Ω = B δ ( x ε , j ) with δ = 1 4 min h ≠ j ∣ p h − p j ∣ . At first, we compute the left-hand side of (2.5). By the Taylor’s expansion, we have

(3.18) ∫ B δ ( x ε , j ) ∂ V ( y ) ∂ y i u ε v ε d y = ∑ l = 1 3 ∫ B δ ( x ε , j ) ∂ 2 V ( p j ) ∂ y i ∂ y l ( y l − p j , l ) u ε v ε d y + o ∫ B δ ( x ε , j ) ∣ y − p j ∣ u ε v ε d y .

Letting y = ε z + x ε , j , we obtain

(3.19) ∫ B δ ( x ε , j ) ( y l − p j , l ) u ε v ε d y = ε 3 ∫ B δ ε ( 0 ) ( ε z l + x ε , j l − p j , l ) u ε ( ε z + x ε , j ) v ε ( ε z + x ε , j ) d z = ε 4 ∫ B δ ε ( 0 ) z l + x ε , j l − p j , l ε u ε ( ε z + x ε , j ) v ε , j d z .

According to Lemma 3.1 and Proposition 3.2,

∣ x ε , j − p j ∣ = O ( ε 2 ) , ∣ u ε ∣ ≤ C ∑ h = 1 m e − λ ∣ x − x ε , h ∣ ε , x ∈ R 3 ,

thus,

∣ x ε , j l − p j , l ∣ = O ( ε 2 ) , u ε ( ε z + x ε , j ) ≤ C e − λ ∣ z ∣ + ∑ h ≠ j e − λ ∣ ε z + x ε , j − x ε , h ∣ ε .

Therefore, we have

∫ B δ ε ( 0 ) x ε , j l − p j , l ε u ε ( ε z + x ε , j ) v ε , j d z ≤ C ε ‖ v ε , j ‖ L ∞ ( R 3 ) ∫ B δ ε ( 0 ) ∣ u ε ( ε z + x ε , j ) ∣ d z ≤ C ε ‖ v ε , j ‖ L ∞ ( R 3 ) ∫ B δ ε ( 0 ) e − λ ∣ z ∣ + ∑ h ≠ j e − λ ∣ ε z + x ε , j − x ε , h ∣ ε d z .

Since δ = 1 4 min h ≠ j ∣ p h − p j ∣ , we have

(3.20) ∫ B δ ε ( 0 ) x ε , j l − p j , l ε u ε ( ε z + x ε , j ) v ε , j d z ≤ C ε ∫ B δ ε ( 0 ) e − λ ∣ z ∣ + e − λ δ ε d z ≤ C ε .

Combining (3.19) with (3.20), we obtain

(3.21) ∫ B δ ( x ε , j ) ( y l − p j , l ) u ε v ε d y = ε 4 ∫ B δ ε ( 0 ) z l u ε ( ε z + x ε , j ) v ε , j d z + o ( ε 4 ) .

Next, we compute ∫ B δ ε ( 0 ) z l u ε ( ε z + x ε , j ) v ε , j d z . As we know,

u ε ( y ) = ∑ h = 1 m U ε , x ε , h ( y ) + ω ε ( y ) = ∑ h = 1 m ( V ( x ε , h ) ) 1 p − 1 w V ( x ε , h ) y − x ε , h ε + ω ε ( y ) .

By Lemma 3.4, we obtain

(3.22) ∫ B δ ε ( 0 ) z l ω ε ( ε z + x ε , j ) v ε , j d z ≤ ∫ B δ ε ( 0 ) ω ε 2 ( ε z + x ε , j ) d z 1 2 ∫ B δ ε ( 0 ) ∣ z ∣ 2 v ε , j 2 d z 1 2 ≤ C 1 ε 3 ∫ R 3 ω ε 2 ( y ) d y 1 2 ∫ B δ ε ( 0 ) ∣ z ∣ 2 e − σ ∣ z ∣ d z 1 2 ≤ C ε 3 2 ‖ ω ε ‖ ε = o ( 1 )

and

(3.23) ∫ B δ ε ( 0 ) z l ∑ h ≠ j ( V ( x ε , h ) ) 1 p − 1 w V ( x ε , h ) ε z + x ε , j − x ε , h ε v ε , j d z ≤ C ∫ B δ ε ( 0 ) ∣ z ∣ e − V 0 δ ε ∣ v ε , j ∣ d z = o ( 1 ) .

According to (3.21)–(3.23) and Lemma 3.5, we have

∫ B δ ε ( x ε , j ) ( y l − p j , l ) u ε v ε d y = ε 4 ( V ( p j ) ) 1 p − 1 ∫ B δ ε ( 0 ) z l w ( V ( p j ) z ) v ε , j d z + o ( ε 4 ) = ε 4 ( V ( p j ) ) 2 p − 1 a l j ∫ R 3 z l w ( V ( p j ) z ) ∂ w ( V ( p j ) z ) ∂ z l d z + o ( ε 4 ) ,

which, together with (3.18), implies

(3.24) ∫ B δ ( x ε , j ) ∂ V ( y ) ∂ y i u ε v ε d y = ε 4 ( V ( p j ) ) 2 p − 1 ∑ l = 1 3 ∂ 2 V ( p j ) ∂ y i ∂ y l a l j ∫ R 3 z l w ( V ( p j ) z ) ∂ w ( V ( p j ) z ) ∂ z l d z + o ( ε 4 ) .

Next, we estimate the right-hand side of (2.5). Using Lemmas 3.1 and 3.4, there exists γ > 0 such that

(3.25) − ε 2 ∫ ∂ B δ ( x ε , j ) ∂ u ε ∂ n ∂ v ε ∂ y i + ∂ v ε ∂ n ∂ u ε ∂ y i d S + ε 2 ∫ ∂ B δ ( x ε , j ) ∇ u ε ∇ v ε n i d S + ∫ ∂ B δ ( x ε , j ) ( V ( y ) u ε − ∣ u ε ∣ p − 1 u ε ) v ε n i d S = O ( e − γ ε )

and

(3.26) ∫ ∂ B δ ( x ε , j ) Φ u ε u ε v ε n i d S + 1 4 π ∫ ∂ B δ ( x ε , j ) ∫ R 3 u ε ( z ) v ε ( z ) ∣ y − z ∣ d z u ε 2 n i d S = O ( e − γ ε ) .

Denote

∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 u ε 2 ( z ) d z u ε ( y ) v ε ( y ) d y = E 1 + E 2 + E 3 + E 4 + E 5 ,

where

E 1 = ∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 u ε 2 ( z ) d z ω ε ( y ) v ε ( y ) d y ; E 2 = ∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 u ε 2 ( z ) d z ∑ h ≠ j U ε , x ε , h ( y ) v ε ( y ) d y ; E 3 = ∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 u ε ( z ) ∑ h ≠ j U ε , x ε , h ( z ) + ω ε ( z ) d z U ε , x ε , j ( y ) v ε ( y ) d y ; E 4 = ∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 U ε , x ε , j 2 ( z ) d z U ε , x ε , j ( y ) v ε ( y ) d y ; E 5 = ∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 U ε , x ε , j ( z ) ∑ h ≠ j U ε , x ε , h ( z ) + ω ε ( z ) d z U ε , x ε , j ( y ) v ε ( y ) d y .

By Hardy-Littlewood-Sobolev inequality, Hölder inequality, Theorem 1.1, and Lemma 3.3, we have

∣ E 1 ∣ ≤ ‖ u ε ‖ L 3 ( R 3 ) 2 ‖ ω ε ‖ L 3 ( R 3 ) ‖ v ε ‖ L 3 ( R 3 ) = O ( ε 5 )

and

∣ E 5 ∣ ≤ ∑ h ≠ j U ε , x ε , j U ε , x ε , h L 3 2 ( R 3 ) + ‖ U ε , x ε , j ω ε ‖ L 3 2 ( R 3 ) ‖ U ε , x ε , j v ε ‖ L 3 2 ( R 3 ) ≤ ∑ h ≠ j U ε , x ε , j U ε , x ε , h L 3 2 ( R 3 ) + ‖ U ε , x ε , j ‖ L 3 ( R 3 ) ‖ ω ε ‖ L 3 ( R 3 ) ‖ U ε , x ε , j ‖ L 3 ( R 3 ) ‖ v ε ‖ L 3 ( R 3 ) = O ε 4 ∑ h ≠ j e − V 0 2 ε ∣ x ε , j − x ε , h ∣ + ε = O ( ε 5 ) .

Similarly, we have

∣ E 2 ∣ = ε 3 ∫ B δ ( x ε , j ) ∫ R 3 ε x i + x ε , j i − y i ∣ y − ( ε x + x ε , j ) ∣ 3 u ε 2 ( ε x + x ε , j ) d x ∑ h ≠ j U ε , x ε , h ( y ) v ε ( y ) d y = ε 6 ∫ B δ ε ( 0 ) ∫ R 3 ε x i + x ε , j i − ( ε z i + x ε , j i ) ∣ ( ε z + x ε , j ) − ( ε x + x ε , j ) ∣ 3 u ε 2 ( ε x + x ε , j ) d x ∑ h ≠ j U ε , x ε , h ( ε z + x ε , j ) v ε , j ( z ) d z = O e − γ ε .

By utilizing Lemma 3.5 and the symmetry of w , we obtain

∣ E 4 ∣ = ε 4 ( V ( p j ) ) 3 p − 1 ∫ B δ ε ( 0 ) ∫ R 3 z i − y i ∣ y − z ∣ 3 w 2 ( V ( p j ) z ) w ( V ( p j ) y ) v ε , j ( y ) d z d y + o ( ε 4 ) = ε 4 ( V ( p j ) ) 4 p − 1 ∑ h = 1 3 a h j ∫ B δ ε ( 0 ) ∫ R 3 z i − y i ∣ y − z ∣ 3 w 2 ( V ( p j ) z ) w ( V ( p j ) y ) ∂ w ( V ( p j ) y ) ∂ y h d z d y + o ( ε 4 ) = ε 4 ( V ( p j ) ) 4 p − 1 a i j ∫ B δ ε ( 0 ) ∫ R 3 z i − y i ∣ y − z ∣ 3 w 2 ( V ( p j ) z ) w ( V ( p j ) y ) ∂ w ( V ( p j ) y ) ∂ y i d z d y + o ( ε 4 ) .

Next, we estimate E 3 . Expand E 3 as follows:

E 3 = E 31 + E 32 + E 33 + E 34 ,

where

E 31 = ∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 u ε ( z ) ω ε ( z ) U ε , x ε , j ( y ) v ε ( y ) d z d y ; E 32 = ∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 U ε , x ε , j ( z ) ∑ h ≠ j U ε , x ε , h ( z ) U ε , x ε , j ( y ) v ε ( y ) d z d y ; E 33 = ∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 ∑ h ≠ j U ε , x ε , h ( z ) 2 U ε , x ε , j ( y ) v ε ( y ) d z d y ; E 34 = ∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 ω ε ( z ) ∑ h ≠ j U ε , x ε , h ( z ) U ε , x ε , j ( y ) v ε ( y ) d z d y .

Similarly, by using Hardy-Littlewood-Sobolev inequality, Hölder inequality, Theorem 1.1, and Lemma 3.3, we obtain

E 31 = O ( ε 5 ) , E 32 = O e − γ ε , E 34 = O ( ε 5 ) .

By Corollary 2.3 and Fubini theorem, we have

∣ E 33 ∣ ≤ ∑ h ≠ j ∫ B δ ( x ε , j ) ∫ R 3 1 ∣ y − z ∣ 2 U ε , x ε , h 2 ( z ) U ε , x ε , j ( y ) d z d y + O e − γ ε = ∑ h ≠ j ε ∫ B δ ε ( 0 ) ∫ R 3 1 ∣ x − z − x ε , j ε ∣ 2 U ε , x ε , h 2 ( z ) ( V ( x ε , j ) ) 1 p − 1 w ( V ( x ε , j ) x ) d z d x + O e − γ ε ≤ C ε 3 ∑ h ≠ j ∫ R 3 1 ∣ z − x ε , j ∣ 2 U ε , x ε , h 2 ( z ) d z + O e − γ ε = C ε 6 ( V ( x ε , h ) ) 2 p − 1 ∑ h ≠ j ∫ R 3 1 ∣ ε y + x ε , h − x ε , j ∣ 2 w 2 ( V ( x ε , h ) y ) d y + O e − γ ε ≤ C ε 6 ∑ h ≠ j 1 ∣ x ε , h − x ε , j ∣ 2 + O e − γ ε = O ( ε 6 ) .

Thus,

(3.27) ∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 u ε 2 ( z ) d z u ε ( y ) v ε ( y ) d y = ε 4 ( V ( p j ) ) 4 p − 1 a i j ∫ B δ ε ( 0 ) ∫ R 3 z i − y i ∣ y − z ∣ 3 w 2 ( V ( p j ) z ) w ( V ( p j ) y ) ∂ w ( V ( p j ) y ) ∂ y i d z d y + o ( ε 4 ) = ε 4 ( V ( p j ) ) 4 p − 1 a i j ∫ R 3 ∫ R 3 z i − y i ∣ y − z ∣ 3 w 2 ( V ( p j ) z ) w ( V ( p j ) y ) ∂ w ( V ( p j ) y ) ∂ y i d z d y + o ( ε 4 ) .

Analogous to (3.27), we have

(3.28) ∫ B δ ( x ε , j ) ∫ R 3 z i − y i ∣ y − z ∣ 3 u ε ( z ) v ε ( z ) d z u ε 2 ( y ) d y = ε 4 ( V ( p j ) ) 4 p − 1 a i j ∫ R 3 ∫ R 3 y i − z i ∣ y − z ∣ 3 w 2 ( V ( p j ) z ) w ( V ( p j ) y ) ∂ w ( V ( p j ) y ) ∂ y i d z d y + o ( ε 4 ) .

Therefore, it follows from (2.5) and (3.25)–(3.28) that

∫ B δ ( x ε , j ) ∂ V ( y ) ∂ y i u ε v ε d y = o ( ε 4 ) ,

which, together with (3.24), gives

( D 2 V ( p j ) ) 3 × 3 a 1 j a 2 j a 3 j = o ( 1 ) .

By ( V 2 ) , we have

□ a i j = 0 , i = 1 , 2 , 3 ; j = 1 , 2 , … , m .

Proof of Theorem 1.3

By Lemma 3.4, there exists R > 0 such that

(3.29) ∣ v ε ( y ) ∣ ≤ C e − σ R < 1 2 , y ∈ R 3 \ ⋃ j = 1 m B ε R ( x ε , j ) .

According to Lemmas 3.5 and 3.6, we have

v ε , j → 0 in C 1 ( B R ( 0 ) ) as ε → 0 ,

which implies

(3.30) v ε ( y ) = o ( 1 ) , y ∈ B ε R ( x ε , j ) , j = 1 , 2 , … , m .

It follows from (3.29) and (3.30) that

∣ v ε ( y ) ∣ < 1 2 , y ∈ R 3 ,

which is a contradiction to ‖ v ε ‖ L ∞ ( R 3 ) = 1 . Therefore, v ε = 0 for small ε .□

Acknowledgements

The authors would like to acknowledge the anonymous referees for valuable comments and suggestions to improve the quality of our manuscript.

Funding information: H.-S. Ding was supported by NSFC and Two Thousand Talents Program of Jiangxi Province (jxsq2019201001). B. Li was supported by NSFC (Nos. 12101274, 12226309), The Jiangxi Province Science Fund for Distinguished Young Scholars (No. 20224ACB218001), and Jiangxi Provincial Department of Education Fund (No. GJJ210336).
Conflict of interest: The authors state no conflict of interest.
Data availability statement: There is no data in this article.

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Received: 2022-11-08

Revised: 2023-05-26

Accepted: 2023-05-28

Published Online: 2023-07-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/ans-2022-0079

Keywords for this article

multi-peak solution; non-degeneracy; Schrödinger-Poisson system

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